Aaron Doherty, 24, also known as Harry, was returning home on July 12 When he saw the incident in the Skeoge area of the city, his family said.
He was assaulted as he tried to intervene.
Aaron’s stepfather Gerry Doherty said the family has been “living a nightmare” since the attack.
“Because of these guys attacking this dog, he intervened. But he has ended up in a nightmare situation himself and indeed we are in the nightmare with him,” he told BBC Radio Foyle.
Aaron had been travelling home in a taxi when he saw a gang of about 15 young people throwing “sticks and bottles” at a black dog, his stepfather said.
He said Aaron would have felt compelled to intervene.
“He is a very passionate guy about animals and particularly about dogs.
“Harry is the type of guy who would be walking along and if a dog was in the river he would jump in, he wouldn’t think twice and he wouldn’t be a great swimmer.
“It is an instinct in him”.
Aaron asked the taxi driver to stop, his stepfather said, and he got out and asked “the boys what they were doing”.
“We don’t know if he was punched or if he was hit with something but he sustained an injury to his head and fell to the ground,” Mr Doherty said.
The gang ran off, but two passers-by stayed with Aaron until an ambulance arrived.
“Without them things could have been a lot worse.”
Aaron was initially treated at Derry’s Altnagelvin Hospital, but was transferred to Belfast’s Royal Victoria on Tuesday.
His family is appealing to anyone involved, or to those who know what happened, to come forward.
Mr Doherty is encouraging the “good young people” who may have been among the gang who may be too afraid to speak out, to tell someone what happened.
“Give yourself up, go to a priest, go to the police, go anywhere, but give yourself up.
“I would be fearful that the person that done this could do it again with more serious implications down the line,” he added.
Police said investigations are ongoing into the attack.
They have appealed for anyone with information to come forward and contact detectives at Strand Road on the 101 number.
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